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In this paper, size effects exhibited by mechanical metamaterials have been studied. When the sizescale of the metamaterials is reduced, stiffening or softening responses are observed in experiments. In order to capture both the stiffening and softening size effects fully, a second-order asymptotic homogenization method based on strain gradient theory is used. By this method, the metamaterials are homogenized and become effective strain gradient continua. The effective metamaterial parameters including the classical and strain gradient stiffness tensors are calculated. Comparisons between a detailed finite element analysis and the effective strain gradient continua model have been made for metamaterials under different boundary conditions, different aspect ratios, different unit cells (closed or open cells) and different topologies. It shows that both stiffening and softening size effects can be captured by using the effective strain gradient continua models.

Size effects of mechanical metamaterials: a computational study based on a second-order asymptotic homogenization method